Abstract

Subpixel methods that locate curves and their singularities, and that accurately measure geometric quantities, such as orientation and curvature, are of significant importance in computer vision and other applications. Such methods often use local surface fits or structural models for a local neighborhood of the curve, to obtain the interpolated curve. Whereas their performance is good in smooth regions of the curve, it is typically poor in the vicinity of singularities. Similarly, the computation of geometric quantities is often regularized to deal with noise present in discrete data. However, in the process, discontinuities are blurred over, leading to poor estimates at them, and in their vicinity. In this paper we propose a geometric interpolation technique to overcome these limitations by locating curves and obtaining geometric estimates while: 1) not blurring across discontinuities, and 2) explicitly and accurately placing them. The essential idea is to avoid the propagation of information across singularities. This is accomplished by a one-sided smoothing technique, where information is propagated from the direction of the side with the ``smoother'' neighborhood. When both sides are non-smooth, the two existing discontinuities are relieved by placing a single discontinuity, or shock. The placement of shocks is guided by geometric continuity constraints, resulting in subpixel interpolation with accurate geometric estimates. The interpolations are shown to be better than spline-like interpolations in smooth regions, and far better in discontinuous ones. Since the technique was originally motivated by curve evolution applications, we demonstrate its usefulness in capturing not only smooth evolving curves, but also discontinuous ones. In particular, the technique is shown to be far better than traditional methods when multiple or entire curves are present in a very small neighborhood.